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Thread: Perimeter inequality

  1. #1

    Perimeter inequality

    We have a triangle with sides a, b, c that ab+bc+ca=12
    Prove that 6=<a+b+c<7
    I proved that a+b+c is no less than 6 but how to prove that it's less than 7?

  2. #2
    Elite Member
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    Quote Originally Posted by eligotas View Post
    We have a triangle with sides a, b, c that ab+bc+ca=12
    Prove that 6=<a+b+c<7
    I proved that a+b+c is no less than 6 but how to prove that it's less than 7?
    How did you prove that (a+b+c) is greater than 6?

    Please share your work - indicating exactly where you are stuck.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
    Elite Member
    Join Date
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    7,618
    Quote Originally Posted by eligotas View Post
    We have a triangle with sides a, b, c that ab+bc+ca=12
    Prove that 6=<a+b+c<7
    I proved that a+b+c is no less than 6 but how to prove that it's less than 7?
    Quote Originally Posted by Subhotosh Khan View Post
    How did you prove that (a+b+c) is greater than 6?
    Please share your work - indicating exactly where you are stuck.
    TO eligotas, If you will answer Prof Khan's question then I will show why the upper bound.
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

  4. #4
    a2+b2 >= 2ab
    a2+c2 >= 2ac
    b2+c2 >= 2bc

    a2+b2+c2 >= ab+ac+bc
    a2+b2+c2+2ab+2ac+2bc >= 3ab+3ac+3bc
    (a+b+c)2 >= 36
    a+b+c >=6
    Last edited by eligotas; 01-15-2019 at 10:05 AM.

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